Applications of homotopy theory to 4D geometry, number theory, and physics

同伦理论在最近的四维几何、物理和数论的研究中扮演着重要的角色：它是一种技术上强大的意识形态，它揭示了表面上看起来不相关的主题之间的深刻联系。Madsen, Tillmann, Weiss， Cohen等人最近关于黎曼曲面模空间的稳定上同调的研究是基于弦物理思想中某些协同范畴的研究。这些结构具有四维的类似物（在物理学中具有类似的意义），这与哈彻、瓦尔德豪森等人在80年代和90年代提出的伪同位素理论有着惊人的联系。这个理论以一种非平凡的方式涉及到整数的代数k理论；后一个主题现在被认为在一个显然不相关的思想圈中发挥重要作用，将有理数域的绝对伽罗瓦群与代数几何中的动机理论联系起来（通过Deligne， Goncharov， Kontsevich等人的工作）。这些想法反过来又涉及到Connes， Kreimer和其他人对物理学中经典重整化理论的重新解释。这个提议的中心概念之一是希望在微分拓扑中找到代数几何家混合泰特动机的类似物，这将与整数的代数k理论有关，就像这个主题与瓦尔德豪森的空间代数k理论有关一样。用不那么专业的术语来说：同伦理论为平等地研究数学对象及其变形提供了一套技术；事实上，研究变形的变形也是同样快乐的，等等。这一提议最终涉及两套根植于物理学的思想，一套来自现代弦理论，另一套来自更经典的重整化理论。前一组思想与微分拓扑有很深的联系，后一组思想则与代数几何和数论的最新发展有关。从表面上看，微分拓扑与算术代数几何相距甚远，但它们通过整数的代数k理论联系在一起，k理论是同伦理论的一部分。因为这门学科系统地处理一种理论到另一种理论的变形，它提供了一套非常方便的工具来联系这些不同的学科。这个建议提出了一种澄清这些联系的方法，基于空间的k理论而不是数字的思想，这是由瓦尔德豪森首先提出的，最近被算术几何领域的工作者（如Soul\'e）进一步推广。这个项目将以一种概念上令人满意的方式把这些重要的数学和物理学的最新发展结合在一起。

DMS-0406461 Jack MoravaHomotopy theory plays an important role in recent work in four-dimensional geometry, physics, and number theory: it is a technically powerful ideology, which exposes deep connections among topics which may on the surface seem unrelated. Recent work of Madsen, Tillmann, Weiss, Cohen, and others on the stable cohomology of the moduli space of Riemann surfaces is based on the study of certain cobordism categories suggested by ideas from string physics. These constructions have four-dimensional analogs (of similar significance in physics), which display surprising connections to the pseudoisotopy theory developed in the 80's and 90's by Hatcher, Waldhausen, and others. That theory involves the algebraic K-theory of the integers in a nontrivial way; and the latter subject is now seen to play an important role in an apparently unrelated circle of ideas connecting the absolute Galois group of the rational number field to the theory of motives in algebraic geometry (through work of Deligne, Goncharov, Kontsevich, and others). These ideas are in turn involved in work of Connes, Kreimer, and others on a reinterpretation of the classical theory of renormalization in physics. One of the central notions of this proposal is the hope of finding in differential topology an analog of the algebraic geometers' mixed Tate motives, which would be related to the algebraic K-theory of the integers as that subject is to Waldhausen's algebraic K-theory of spaces.In less technical terms: homotopy theory provides a body of techniques for studying mathematical objects and their deformations on an equal footing; indeed, it is equally happy studying deformations of deformations, and so on. This proposal is concerned ultimately with two sets of ideas with roots in physics, one coming from modern string theory, the other from the more classical theory of renormalization. The former set of ideas has deep connections with differential topology, and the latter is related to recent developments in algebraic geometry and number theory. On the surface, differential topology and arithmetic algebraic geometry are far apart, but they are linked through the algebraic K-theory of the integers, which is at base a part of homotopy theory. [Because that subject deals so systematically with deformations of one theory into another, it provides a very convenient set of tools for relating such disparate subjects.] This proposal suggests a way of clarifying these linkages, based on ideas about the K-theory of spaces rather than numbers, which were first proposed by Waldhausen, and which have lately given further currency by workers in arithmetic geometry such as Soul\'e. This program would bring together these important recent developments in mathematics and physics in a conceptually satisfying way.